Math 535 - General Topology Additional notes

Martin Frankland

November 12, 2012

1 Partitions of unity

Definition 1.1. Let X be a topological space and f : X → R a continuous function. The support of f is the closed subset

supp f := {x ∈ X | f(x) 6= 0} ⊆ X.

Definition 1.2. A cover {Uα}α∈A of a space X is locally finite if for all x ∈ X, there is a neighborhood Nx of x that intersects only finitely many of the Uα, i.e.

Nx ∩ Uα 6= ∅ for finitely many α ∈ A. Note that the definition applies to any kind of cover, not just open covers.

Definition 1.3. A partition of unity on a space X is a family of continuous functions {ρβ : X → [0, 1]}β∈B satisfying the following two properties.

1. The family {supp ρβ}β∈B is locally finite. P 2. β∈B ρβ(x) = 1 for all x ∈ X.

In particular, the supports {supp ρβ}β∈B form a cover of X, and for all x ∈ X, ρβ(x) = 0 for all except finitely many indices β ∈ B.

The partition of unity is subordinate to a cover {Uα}α∈A of X if for all β ∈ B, there is an index α = α(β) ∈ A satisfying supp ρβ ⊆ Uα(β). Remark 1.4. One could assume WLOG that the indexing set B is A. Indeed, one could add together the functions ρβ supported on the same Uα, and consider the constant zero function for each index α that does not appear as α(β).

It is useful to have partitions of unity subordinate to any open cover. We will find conditions on a space that make this always possible.

1 2 Paracompactness

Definition 2.1. Let U be a cover of X.A refinement of U is a cover V of X such that every member V ∈ V is a subset of some U ∈ U.

In other words, a cover {Vβ}β∈B is a refinement of the cover {Uα}α∈A if for all β ∈ B, there is an index α(β) ∈ A satisfying Vβ ⊆ Uα(β).

Example 2.2. {(n, n + 2)}n∈Z is a refinement of the cover {R} of R. ∞ Example 2.3. {(0, ∞)} is a refinement of the cover {(n, ∞)}n=0 of (0, ∞). Definition 2.4. A topological space X is paracompact if every open cover of X admits a locally finite open refinement.

Remark 2.5. Some authors (e.g. Bredon, Willard) include the Hausdorff condition in the defi- nition of paracompact. Other authors (e.g. Munkres, Wikipedia) do not assume that the space is Hausdorff. We will not assume that the definition of paracompact includes Hausdorff. Example 2.6. Every compact space is paracompact. Example 2.7. Rn is paracompact. This will follow from 2.13. Definition 2.8. A space is σ-compact if it is a countable union of compact subspaces.

Example 2.9. n is σ-compact, as it is the union of closed balls n = S B (0). R R k∈N k Example 2.10. Any closed subset of Rn is σ-compact. Example 2.11. More generally, any closed subset C ⊆ X of a σ-compact Hausdorff space X is σ-compact. Example 2.12. Any second-countable manifold is σ-compact. More generally, any second- countable locally compact space is σ-compact (c.f. Homework 12 #2).

Proposition 2.13. Every locally compact, σ-compact, Hausdorff space is paracompact.

Proposition 2.14. A closed subspace of a paracompact space is paracompact.

Proof. Homework 12 #3.

Lemma 2.15. If {Ai}i∈I is a locally finite collection of subsets Ai ⊆ X, then we have [ [ Ai = Ai. i∈I i∈I Proposition 2.16. Every paracompact Hausdorff space is normal.

Proposition 2.17 (Shrinking lemma). Let X be a paracompact Hausdorff space and {Uα}α∈A an open cover. Then there is a locally finite open cover {Vα}α∈A satisfying Vα ⊆ Uα for all α ∈ A.

Note that the indexing set A is the same for both open covers. Note also that Vα is allowed to be empty.

Theorem 2.18 (Existence of partitions of unity). Let X be a paracompact Hausdorff space and U = {Uα}α∈A an open cover. Then there is a partition of unity on X subordinate to U.

2 3 Applications to manifolds

Theorem 3.1. Any compact Hausdorff manifold can be embedded in RN for some N.

Proof. Let M be an m-dimensional compact Hausdorff manifold. For each x ∈ M, a coordinate m chart about x consists of a homeomorphism ϕx : Ux → Vx ⊆ R where Ux is an open neigh- m borhood of x and Vx is an open subset of R . Since M is compact, the open cover {Ux}x∈M admits a finite subcover {U1,...,Uk}. Since M is paracompact and Hausdorff, there exists a k partition of unity {ρi}i=1 with supp ρi ⊆ Ui. m For i = 1, . . . , k, define maps hi : M → R by ( ρi(x)ϕi(x) if x ∈ Ui hi(x) = 0 otherwise.

These maps hi are well defined and continuous. Now define the continuous map

k k z }| { z m }| m{ ∼ k(m+1) g : M → R × ... × R × R × ... × R = R

x 7→ (ρ1(x), . . . , ρk(x), h1(x), . . . , hk(x)) .

Since M is compact and RN is Hausdorff, g is a closed map. To show that g is an embedding, it remains to show that g is injective.

Assume g(x) = g(y) for some x, y ∈ M. Then ρi(x) = ρi(y) for all i. Let j be an index where ρj(x) > 0 and thus ρj(y) = ρj(x) > 0. We obtain

hj(x) = hj(y)

ρj(x)ϕj(x) = ρj(y)ϕj(y)

⇒ ϕj(x) = ϕj(y) ⇒ x = y since ϕj is injective.

This example illustrates how partitions of unity on a manifold can be useful. Since many interesting manifolds are not compact, it would be useful to know when a (Hausdorff) manifold is paracompact.

Theorem 3.2. Let M be a Hausdorff manifold. Then M is paracompact if and only if each connected component of M is second-countable.

Proof. Let {Mi}i∈I be the connected components of M. Recall that manifolds are locally path- ` connected. Therefore M = i∈I Mi is the coproduct of its connected components (which are the same as its path components).

(⇐) Note that any manifold is locally compact. Since Mi is locally compact and second- countable, it is σ-compact (by 2.12).

Since Mi is locally compact, σ-compact, and Hausdorff, it is paracompact (by 2.13). Since ` M = i∈I Mi is a coproduct of paracompact spaces, it is paracompact (c.f. Homework 12 #4).

3 (⇒) Since Mi is connected, locally compact, paracompact, and Hausdorff, it is σ-compact (by 3.3).

Since Mi is σ-compact and every point x ∈ Mi has a second-countable neighborhood, Mi is second-countable (by 3.4).

Proposition 3.3. Let X be a connected, locally compact, paracompact, Hausdorff space. Then X is σ-compact.

Proposition 3.4. Let X be a σ-compact space such that every point x ∈ X has a second- countable neighborhood. Then X is second-countable.

Here is another application of partitions of unity.

Proposition 3.5. Every paracompact smooth manifold admits a Riemannian metric.

Yet another application is in defining integration on manifolds. One can define integration within a coordinate chart, and then on the entire manifold using a partition of unity.

4